2. Objectives:
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β Illustrate the different types of conic sections β parabola,
hyperbola, ellipse, circle, and degenerate cases;
β Define a circle;
β Determine the standard from of equation of a circle; and
β Graph a circle in the rectangular coordinate plane.
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
4. The Origin
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TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
Apollonius of Perga
β Greek Geometer
β Studies the curves formed
by the intersection of a
plane and a double right
circular cone.
5. The βCone-icβ Sections
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TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
β Conic section is a curve
formed by the
intersection of a plane
and a double right circular
cone.
7. The Three Conic Sections
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β Parabola is formed if the cutting plane is parallel to one and
only one generator
β Ellipse is formed if the cutting of the plane is not parallel to
any generator.
β Circle is formed if the cutting of the plane is not parallel
to any to any generator but is perpendicular to the axis.
β Hyperbola is formed if the cutting plane is parallel to two
generators.
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
8. The Three Conic Sections
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TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
10. Conic Sections as Defined
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β A conic is a set of points
whose distances from a
fixed point are in constant
ratio to their distances
from a fixed line that is not
passing through the fixed
point.
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
11. Elements
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Focus (F) β the fixed point of a conic
Directrix (d) β the fixed line d
corresponding to the focus.
Principal axis (a) β a line that
passes through the focus and
perpendicular to the directrix. Every
conic is symmetric with respect to its
principal axis.
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
12. Elements
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Vertex (V) β the point of intersection of
the conic and its principal axis
Eccentricity (e) β the constant ratio. If
point P is one of the points of the conic
with point Q as its projection on d, then
the eccentricity is the ratio of the
distance |FP| to the distance |QP|, which
is a constant. In symbols,
π =
|ππ·|
|πΈπ·|
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
13. Eccentricity
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β The conic is a parabola if the
eccentricity π = 1.
β The conic is an ellipse if the
eccentricity π < 1.
β The conic is circle if the
eccentricity π = 0.
β The conic is a hyperbola if the
eccentricity π > 1.
TSU β LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
15. Circle Defined
A circle is a set of all coplanar
points such that the distance from
a fixed point is constant. The fixed
point is called the center of the
circle and the constant distance
from the center is called the
radius of the circle.
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16. Circle Defined
A circle is a set of all coplanar points
such that the distance from a fixed
point is constant. The fixed point is
called the center of the circle and
the constant distance from the
center is called the radius of the
circle.
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17. Equation of a Circle with C (0, 0)
π₯2
+ π¦2
= π2
This equation is referred to as the
standard form of equation of a circle
whose center is at the origin with
radius r.
This can be derived using the
distance formula.
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19. Equation of a Circle with C (h,k)
(π₯ β β)2
+(π¦ β π)2
= π2
This equation of the circle whose
center is at the point (β, π) and with
radius π
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20. Exercise
Determine the standard form of equation of the following circles
and the radius. Draw the graph.
1. Center πΆ(0, 0), radius: 9;
2. Center πΆ(β5,7), radius: 6;
3. Center πΆ( 7, 3 3), radius: 8.
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24. Equation of a Circle in GF
ππ + ππ + π«π + π¬π + π = π,
where π· = β2β, E = β2k, and F = β2
+ π2
β π2
This general form was derived by expanding the standard form of
the equation.
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25. Derivation
ππ + ππ + π«π + π¬π + π = π,
where π· = β2β, E = β2k, and F = β2
+ π2
β π2
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(π₯ β β)2
+(π¦ β π)2
= π2
29. Examples
Determine the center and radius of the circle in general form.
π₯2 + π¦2 β 4π₯ β 2π¦ β 4 = 0
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30. Examples
Determine the center and radius of the circle in general form.
π₯2 + π¦2 β 10π₯ β 6π¦ β 18 = 0
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31. Examples
Determine the center and radius of the circle in general form.
2π₯2 + 2π¦2 β 16π₯ + 12π¦ β 4 = 0
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32. Equation of a Circle in GF
ππ + ππ + π«π + π¬π + π = π,
where π· = β2β, E = β2k, and F = β2
+ π2
β π2
Solving for r in terms of D, E, and F in the general form:
If
π·2
4
+
πΈ2
4
β πΉ > 0, then the graph of the equation is a circle.
If
π·2
4
+
πΈ2
4
β πΉ = 0, then the graph of the equation is a point circle.
If
π·2
4
+
πΈ2
4
β πΉ < 0, the equation has no graph.
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33. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
π₯2 + π¦2 + 8π₯ + 15 = 0
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34. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
π₯2 + π¦2 + 87π₯ + 5π¦ + 16 = 0
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35. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
π₯2 + π¦2 + 2π₯ β 6π¦ + 12 = 0
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36. Examples
Find the general equation of a circle.
The center of the circle is at ( -3, 7) and goes through the origin.
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37. Examples
Find the general equation of a circle.
The center of the circle is at (7, 4) and goes through the point (4, 8)
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38. Examples
Find the equation of a circle
The circle passes through the origin, and contain the points (0, 5), and (3, 3).
π₯2
+ π¦2
+ π·π₯ + πΈπ¦ + πΉ = 0
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39. Examples
Find the equation of a circle
The circle passes through the origin, and contain the points (0, 8), and (5, 5).
π₯2
+ π¦2
+ π·π₯ + πΈπ¦ + πΉ = 0
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40. Examples
Find the equation of a circle
The circle passes through the points (2, 3), (6, 1), and (4, -3).
π₯2
+ π¦2
+ π·π₯ + πΈπ¦ + πΉ = 0
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41. Examples
A particular cellphone tower is designed to service a 15-km radius. The tower is located at
(5, -3) on a coordinate plane whose units represents km.
β What is the standard form equation of the outer boundary of the region serviced by
the tower?
β What is the general form equation of the region serviced by the tower?
β Draw the graph of the region serviced by the tower.
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